"inverse semigroup" meaning in English

{
  "forms": [
    {
      "form": "inverse semigroups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "inverse semigroup (plural inverse semigroups)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
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          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
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            "Fundamental"
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        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Group theory",
          "orig": "en:Group theory",
          "parents": [
            "Algebra",
            "Mathematics",
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            "All topics",
            "Fundamental"
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        }
      ],
      "examples": [
        {
          "ref": "2002, Peter G. Trotter, “A.9: Regular Semigroups”, in Aleksandr Vasilʹevich Mikhalev, Günter Pilz, editors, The Concise Handbook of Algebra, page 35",
          "text": "Examples of regular semigroups include any band, inverse semigroup or completely regular semigroup (see sections A. 2, A. 11, and A. 10); in particular, any group is a regular semigroup.",
          "type": "quotation"
        },
        {
          "ref": "2008, Olexandr Ganyushkin, Volodymyr Mazorchuk, Classical Finite Transformation Semigroups: An Introduction, page v",
          "text": "Inverse semigroups form a class of semigroups which are closest (in some sense) to groups.",
          "type": "quotation"
        },
        {
          "ref": "2014, Christopher Hollings, Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups, page 249",
          "text": "Inverse semigroups are central to modern semigroup theory: arguably, they form the most-studied class of semigroups.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A semigroup in which every element x has an inverse y, such that x = xyx and y = yxy."
      ],
      "id": "en-inverse_semigroup-en-noun-yBYz9z4i",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "group theory",
          "group theory"
        ],
        [
          "semigroup",
          "semigroup"
        ],
        [
          "inverse",
          "inverse"
        ]
      ],
      "raw_glosses": [
        "(algebra, group theory) A semigroup in which every element x has an inverse y, such that x = xyx and y = yxy."
      ],
      "related": [
        {
          "word": "regular semigroup"
        }
      ],
      "topics": [
        "algebra",
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "de",
          "lang": "German",
          "sense": "semigroup in which every element has an inverse",
          "tags": [
            "feminine"
          ],
          "word": "inverse Halbgruppe"
        }
      ]
    }
  ],
  "word": "inverse semigroup"
}

{
  "forms": [
    {
      "form": "inverse semigroups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
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      "expansion": "inverse semigroup (plural inverse semigroups)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "regular semigroup"
    }
  ],
  "senses": [
    {
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        "English countable nouns",
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        "en:Algebra",
        "en:Group theory"
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      "examples": [
        {
          "ref": "2002, Peter G. Trotter, “A.9: Regular Semigroups”, in Aleksandr Vasilʹevich Mikhalev, Günter Pilz, editors, The Concise Handbook of Algebra, page 35",
          "text": "Examples of regular semigroups include any band, inverse semigroup or completely regular semigroup (see sections A. 2, A. 11, and A. 10); in particular, any group is a regular semigroup.",
          "type": "quotation"
        },
        {
          "ref": "2008, Olexandr Ganyushkin, Volodymyr Mazorchuk, Classical Finite Transformation Semigroups: An Introduction, page v",
          "text": "Inverse semigroups form a class of semigroups which are closest (in some sense) to groups.",
          "type": "quotation"
        },
        {
          "ref": "2014, Christopher Hollings, Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups, page 249",
          "text": "Inverse semigroups are central to modern semigroup theory: arguably, they form the most-studied class of semigroups.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A semigroup in which every element x has an inverse y, such that x = xyx and y = yxy."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "group theory",
          "group theory"
        ],
        [
          "semigroup",
          "semigroup"
        ],
        [
          "inverse",
          "inverse"
        ]
      ],
      "raw_glosses": [
        "(algebra, group theory) A semigroup in which every element x has an inverse y, such that x = xyx and y = yxy."
      ],
      "topics": [
        "algebra",
        "group-theory",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "translations": [
    {
      "code": "de",
      "lang": "German",
      "sense": "semigroup in which every element has an inverse",
      "tags": [
        "feminine"
      ],
      "word": "inverse Halbgruppe"
    }
  ],
  "word": "inverse semigroup"
}